1. Field of the Invention
This invention relates generally to means and methods for determining Moments of Inertia and, more particularly, relates to means and graphical methods of calculating Moments of Inertia of an irregular shaped area with respect to a fixed axis using specially designed graph paper that enables one to distort the "irregular shaped area" in such a way that the "area of the distorted figure" equals in numerical value the Moment of Inertia of the original "irregular shaped area" about its selected axis of rotation.
2. Description of the Prior Art
One definition of the Moments of Inertia of an area is defined as the sum of the products of each small element of area multiplied by the square of each element's distance from the axis of rotation. The Moment of Inertia, which is generally represented by "I", varies directly as the area and as the square of the radius of gyration of the area.
Design engineers working in several high technology disciplines have a need to calculate the Moments of Inertia of a variety of irregularly shaped objects such as airfoils, ailerons, rudders and including various hydrodynamic as well as aerodynamic shapes. Algebraic expressions have been derived for computing the Moments of Inertia of common geometric figures such as rectangles and circles. For example, the Moment of Inertia of a rectangle about its base as a neutral axis is:
BD.sup.3 /3 WHERE B IS THE WIDTH OF THE RECTANGLE AND D IS ITS DEPTH OR HEIGHT.
Very few practical Moments of Inertia problems can be solved with simple techniques or algebraic methods and it is usually necessary to perform intricate and drawn-out mathematical computations to arrive at an approximation of the Moments of Inertia of an irregularly shaped area or object about a fixed axis.
Several complicated mechanical integrating machines have been proposed in the prior art for measuring Moments of Inertia. Cost and/or application limitations have precluded widespread acceptance of these prior art devices.
U.S. Pat. No. 2,535,208 issued Dec. 26, 1950, to R. L. Hoover, the inventor of this application describes a sophisticated Moment of Inertia calculating device which uses 38 movable cards or paper sheets which are fitted into a metal frame. As stated above, the Moment of Inertia of a rectangle with its axis at the base is equal to bd.sup.3 /3.
Using the prior art Hoover Moment of Inertia calculator, an irregular or random area can be drawn on the exposed edges of the cards with the axis at the bottom edge. Inasmuch as the width b did not change and is therefore a constant, it was only necessary to convert or transform the height or depth of the area to d.sup.3 /3 by extending and retracting the cards to the limits of predetermined slots in each card. At the line of unit distance from the axis of interest, where d equals 1, d.sup.3 /3 equals 1/3 and for all distances from the axis, less than unity, d.sup.3 /3 becomes vanishingly small.
To determine the Moment of Inertia of the area drawn on the above cited Hoover calculator, it was only necessary to measure the outlined area of the modified shape with a standard planimeter. However, in practice, lost motion and cumulative clearances in the slots and side flanges proved to be a potential source of error in the Hoover Moment of Inertia calculator.
Accordingly, a need existed for a straightforward simple and more precise means and method of computing Moments of Inertia.